Problem: Determine how many solutions exist for the system of equations. ${-6x+y = -9}$ ${18x-3y = 27}$
Convert both equations to slope-intercept form: ${-6x+y = -9}$ $-6x{+6x} + y = -9{+6x}$ $y = -9+6x$ ${y = 6x-9}$ ${18x-3y = 27}$ $18x{-18x} - 3y = 27{-18x}$ $-3y = 27-18x$ $y = -9+6x$ ${y = 6x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x-9}$ ${y = 6x-9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-6x+y = -9}$ is also a solution of ${18x-3y = 27}$, there are infinitely many solutions.